Find the solution of the equation dy/dx = y^2 - 1.

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Question: Find the solution of the equation dy/dx = y^2 - 1.

Options:

Correct Answer: y = tan(x + C)

Solution:

This is a separable equation. The solution is y = tan(x + C).

Find the solution of the equation dy/dx = y^2 - 1.

Find the solution of the equation dy/dx = y^2 - 1.

Step 1: Identify the equation dy/dx = y^2 - 1. This is a differential equation.
Step 2: Recognize that this is a separable equation, meaning we can separate the variables y and x.
Step 3: Rewrite the equation as dy/(y^2 - 1) = dx. This separates y on one side and x on the other.
Step 4: Factor the left side: y^2 - 1 = (y - 1)(y + 1). So, we have dy/((y - 1)(y + 1)) = dx.
Step 5: Integrate both sides. The left side requires partial fraction decomposition.
Step 6: Set up the partial fractions: 1/((y - 1)(y + 1)) = A/(y - 1) + B/(y + 1). Solve for A and B.
Step 7: After finding A and B, integrate both sides. The left side will give you ln|y - 1| and ln|y + 1|.
Step 8: The right side integrates to x + C, where C is the constant of integration.
Step 9: Combine the results from the integration to form the equation: ln|y - 1| - ln|y + 1| = x + C.
Step 10: Exponentiate both sides to solve for y. This will lead to the solution y = tan(x + C).

Separable Differential Equations – The equation can be separated into functions of y and x, allowing integration on both sides.
Integration Techniques – The solution involves integrating the separated variables to find the general solution.
General Solution and Constants – The solution includes an arbitrary constant (C) that represents the family of solutions.